Following its saturation, the PNS magnetic field may interact with the newly formed disk of fallback matter and is strong enough to influence the fallback accretion mechanism. This interaction was not modelled because our study was focused on the phase of magnetic field amplification. It may nonetheless influence strongly the rotation of the newly-born magnetar. The evolution of the PNS-fallback disk system depends on three characteristic radii [1]}, [2]}, [3]}, [4]}: (i) the magnetospheric radius \(r_{\rm m}\) , which is the radius at which the matter is blocked by the magnetic barrier, (ii) the corotation radius \(r_{\rm c}\)  where the matter has the same rotation frequency as the PNS, and (iii) the light cylinder radius \(r_{\rm lc}\) , which is the ratio of light speed to the PNS rotation frequency. The strong magnetic field repels the magnetosphere behind the corotation radius, i.e. \(r_{\rm c}<r_{\rm m}\) , which stops the accretion and so the PNS spin-up. If the fallback accretion rate of the disk is large enough, the inner part of the disc is still into the light cylinder (i.e. \(r_{\rm lc}>r_{\rm m}\) ) and makes the large-scale magnetic field lines open. The PNS-fallback disk system enters the so-called propeller regime and the PNS angular momentum is transported toward the disk via the magnetic dipole torque. This mechanism is thought to extract the PNS angular momentum very efficiently, e.g. [2]} even predicts magnetars spun down to rotation periods of \(\sim {e6}{s}\)  after \(\sim {e3}{yr}\) . For this reason, this scenario is often invoked to explain the ultra-long period magnetars like 1E 1613 [6]}, [7]}, [8]} or the recently observed GLEAM-X J162759.5-523504.3 [4]}, which have respectively rotation periods of \(\sim {2.4e4}{s}\)  and \(\sim {1.1e3}{s}\) . It would be interesting to include such a spin-down model in our magnetar formation scenario in order to obtain a prediction of the rotation period at later times.
